I was diagnosed with ME and FMS about 10 years ago. I received absolutely the opposite advice. Well. I received a bunch of absolutely worthless advice, amongst which was that I should have kids to keep me company when I'm old. I wish I were joking.
I woke up one morning and realised that there was no effing way I was going to let myself end up in a wheelchair, which was where I was headed. I joined a gym and started very slowly with weights once a week. Then twice a week a few months later Then three times a week. Then I added a small amount of cardio and progressed from there.
Having gone from being bedridden to cycling 20 miles each way on a commute I was told I no longer had ME, may never have had ME, but I might have lupus, only the blood tests were incorrect and yadda yadda yadda.
Whatever. I stopped paying attention. I still get tired -- I don't ever wake up bright-eyed and busy tailed, rather more feeling like I've been run over by a truck, repeatedly -- I have issues with trigger points and arthritis and sun-sensitivity and fatigue. But I've learned to listen to what my body wants and needs and pay attention to it. I balance the need for a minimal amount of exercise to stop me slipping down into that cycle with trying not to do too much because when I over-train it can make me crash for weeks.
The specialists will give you the best advice they can, but it's still your body and you're the one who has to live with it. They don't have a magical connection to see exactly how various activities make you feel. Only you know how you feel. They may offer drugs or cognitive behavioural therapy. They may tell you that some things are advisable and some things aren't. They may tell you some things will help and some things will make it worse. But only you can find out for sure if they're right or not.
I can't advise you on what is best for you. All I can say is that if I'd listened to what my specialist said I'd be on my arse with children crawling around my feet. As it is I'm still cycling and have since started competing in triathlons. I'm not far off 40, so the age difference may be a factor, but you asked if there were any cyclists out there with ME. I know of at least one paralympic athlete who has ME. It isn't necessarily the end of the world. I'd be tempted to ask how much of their advice is coloured by your age.
Do what's right for you.
Sam